In the dynamic analysis of the stator, the vibrational mode shapes were seen to be of the form of axial and circumferential (radial) vibration modes of a cylindrical shell structure.
Figure 4-4 shows the axial vibration modes of the quarter scale wheel structure. To preliminarily verify the numerical results for the vibration mode shapes, a visual correlation of the simulated results is made with those in literature for cylindrical shell structures [46].
By comparison, it is evident that the vibration mode shapes presented in Figure 4-4 and 4-5 accurately depict the axial and radial vibrational modes of a cylindrical shell structure.
High tangential (torque) forces and very high electromagnetic radial forces, both acting in tangential and radial directions respectively are typical for the gap between the rotor and stator of an operating PMSG. The radial force components are by virtue of the permeability interaction of the air gap and the stator iron, and therefore result in the stator vibrating radially with high magnitudes, whereas the tangential force components generate electromagnetic torque on the rotor. To that effect it can be stated that majority of the vibrations within an electrical machine are by virtue of radial and tangential electromagnetic
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forces. Axial forces may exist in an electrical machine. However, they do not contribute to the harmful excitation forces which leads to vibrations and noise in electrical machines.
Even though, there are no known harmful axial excitation forces, it can be seen from Figure 4-4 that, there exist some normal axial vibrations modes for the simulated wheel structure.
Figure 4-4 (a) shows the first axial mode of vibration of the wheel, it is evident that at a frequency of 108.73 Hz, the entire wheel experiences a global bending mode with very high displacement magnitudes. Additionally, as can be seen from Figure 4-4 (b), (c) and (d) if these vibrations modes are not damped out, they may contribute to the wheel having a poor dynamic performance when in operation. Therefore the layered steel sheets in the lightweight wheel structure have been stiffened with polyethylene spacers, whose work is, mainly to damp out any axial vibrations.
Attention is now given to the effect of excitations caused by the interaction between permanent magnets and stator slots (radial electromagnetic forces). In the design of a DD-PMSG, radial vibrations of the stator is the most predominant source of shock and noise.
Illustrated in Figure 4-5 are the circumferential vibration mode shapes for the stator, showing the radial vibration modes of the threaded rods. As can be seen from Figure 4-5 (a), at 301.32 Hz the second radial vibration mode for the stator shows the circumferential section of the wheel structure bowing out in north and south direction and an inverse reciprocal in the west and east circumferential section of the wheel. Additionally, from Figure 4-5 (b), (c), and (d) it is evident that the dynamic performance of the stator will certainly be poor should these radial modes persist when the stator is in operation. In view of this result, it is necessary to now consider the relevant excitation forces (cogging torque and magnetic forces due to magnet to tooth interaction) present, when the synchronous generator is in operation. The reader is reminded that for the discussion on excitation forces and frequency the wheel structure is assumed to have all magnetic laminates in place, which isn’t the case for the quarter-scale wheel used in the vibration testing. Nonetheless, by adding the laminates means the weight of the structure increases, and as stated in equation (2.2-2) the natural frequency of vibration of a structure is dependent on its mass, and that the more mass there is the low the natural vibrating frequency it will have.
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Assuming, the electric generator has the following parameters; a rotational speed of 44 rpm and 120 number of rotor poles, then the electrical frequency [46] for the stator current is 44 Hz. However, if cogging torque is found to be the sixth harmonic frequency, then we have 264 Hz as the cogging torque excitation frequency. Furthermore as the magnets interact with the stator teeth, attracting forces becomes bigger, and since both north and south poles attract each tooth, the frequency doubles and we have 88 Hz being the second excitation frequency.
Now from dynamics point of view, the stator must be designed to operate from about 20%
away from the high magnitude excitation frequency (264 Hz). This is because resonance will occur should the excitation frequency coincide with the stator natural frequency.
Figure 4-7 show the FRF of the radial vibration measurement with velocity magnitude in (m/s). Illustrated in Figure 4-8 is a zoomed FRF. In dictated on the FRF are the first five modes of the wheel radial vibration. The corresponding mode shapes are presented in Figure 4-6 (b) to (f). As can be seen from Figure 4-6 (a) and indicated in Figure 4-8, at a frequency of 264 Hz, the wheel structure vibrates at a magnitude of -89.01 dB, subsequently for modes 1, 2, 3, 4 and 5 (which is -80.37 dB, -75.62 dB, -89.01 dB, -73.32 dB and -78.42 dB, respectively) it is evident that the magnitudes of the first five vibration modes of the wheel supersedes that of the excitation frequency for the entire wheel structure. Furthermore, from Figure 4-7, from 0 to 200 Hz the structure is exceptionally quiet and judging from the magnitude of the peaks between 200 Hz to 266 Hz, it is safe to say that with regards to radial excitation forces, the wheel structure will be able to damp out any global radial vibration modes. The reader is once again reminded that the excitation frequency of 264 Hz is for the case where the magnetic laminates have been installed on the wheel. Now applying the same analogy as discussed earlier, the increase in mass will lead to a decrease in natural frequency.
In other words the stator natural frequencies will then be shifted far away from the excitation frequency.
To verify these results, the predicted natural frequencies obtained from the modal analysis simulation are compared to measured data collected during the experimental modal analysis.
See Table 4-1. Also shown in the table is the percentage difference between the natural frequencies. Figure 4-9 shows a correlation of the last four values of the predicted and measured data with a correlation coefficient cr= 0.995. It is evident from the correlation
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coefficient that the predicted natural frequencies correlates well with the measured natural frequencies.
From Figure 4-6, the dynamic behavior of the wheel structure is seen to be irregular. As a result, the mode shapes of the wheel section do not give a clear representation of the actual modes of vibration. Also, the fact that one was unable to capture the entire structural mode shapes during the vibration measurement contributed to this effect. Furthermore, the level of structural nonlinearity observed in the wheel structure is mostly due to the choice of binding method used on the layered sheet steel elements.
To increase structural damping, plastic ties were used to bind layers together. This relatively loose binding scheme tends to increase frictional effect between mating surfaces, thereby increasing damping. Also, due to the material strength properties of the plastic ties, they could not be tightened beyond their breaking strength. This however, led to the increased interaction of mating surfaces generating friction, which serves as a means to attenuating vibration.
96 4.5 Summary
In the dynamic analysis of the lightweight wheel structure, it was observed that the vibrational mode shapes were similar to that of cylindrical shell structures given in literature.
It was shown that, there exist some normal axial vibrational modes in the numerical analysis of the stator. However, even though there are no (known) axial excitation forces, the layered steel sheets have been stiffened with polymeric spacers, to provide damping of any axial vibrations and shock in the stator.
The stator’s dynamic performance is highly dependent on how well, the dominant excitation forces (cogging torque and magnetic forces due to magnet to tooth interaction) which lead to unwanted vibrations are attenuated. For a worst case scenario (stator without magnetic laminates) it was shown that, harmful vibrations occurred at 270 Hz and above, whereas the approximated excitation frequency is 264 Hz. It is important to emphasize, with regards to equation (2.2-2) that, with added mass (magnetic laminates) the dynamic performance of the quarter-scale wheel structure will be improved based on the fact that, the increased mass will lead to the excitation and stator natural frequency being far apart. This means, the propensity of these two frequencies coinciding to cause resonance will be zero out of one.
The dynamic behavior of the quarter-scale wheel structure was observed to be irregular, and showing nonlinear characteristics. Consequently, the mode shapes, even though was only for a section of the wheel, were observed to be of a peculiar behavior.
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5 CONCLUSIONS
This thesis studied the dynamic modeling of layered steel sheets through the use of finite element method and experimental modal analysis. Subsequently the dynamic analysis of a novel prototype lightweight wheel structure to be used as the stator of an outer rotor direct-drive permanent magnet synchronous generator designed for high power wind turbines is studied based on the developed layered sheet steel model.
The foundation of experimental modal analysis is based on the assumption that the mechanical system or structure is linear, time invariant and obeys Maxwell’s reciprocity.
Accordingly, numerical simulations should emulate as closely as possible, linear and time invariant structures to archive better correlation between simulation and experimental test.
It was shown that for uniformly flat sheet steel or plate elements, the dynamic properties in terms of natural frequency and mode shapes are affected by the choice of element type. A simulation case study comprising six FE models was conducted, to understand the effect of element type and mesh densities on dynamic properties and how these affect computational efficiency of the analysis. Based on the results, it was evident that the natural frequencies for all six FE models differed slightly among each case type, also, it was observed that, finely meshed models, showed a slight drop in frequencies when compared to those with coarse mesh. However, with SHELL181 elements, it is seen that the required CPU time used to solve the model is adequate in terms of efficiency and accuracy.
Furthermore, it was shown that the developed shell modeling technique applied for the layered sheet steel, coupled with a slightly coarse mesh will not only be suitable for implementation into the lightweight wheel structure but also produce accurate results and improve computational efficiency in the entire wheel analysis.
It was observed in the numerical modeling of layered sheet steel elements that, when an even number of stacked layers are analyzed, implementation of the shell modeling technique discussed in this thesis is quite straight forward. However, shell surface contact definitions become challenging when layers are in the order of odd numbers. At which point a 3D solid element is necessary.
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The effect of structural mass and stiffness, on dynamic parameters was observed by comparing the dynamic properties of a 1.25 mm sheet steel element to a 6 mm thick plate. It was shown that, although modes shapes are similar, the natural frequencies of a 1.25 mm sheet steel are lower compared to a 6 mm plate.
Additionally, it was observed, both from experimental and numerical results that, the dynamic parameters for a 5-layer sheet steel bound with plastic ties, correlates very well with a 1.25 mm sheet steel element. The interaction of the mating interfaces for these layered sheet steel elements suggests that, at each mode of vibration, only the mass of a single sheet steel element is accounted for, and that, it’s as if the mass of the other four sheet steel elements are inactive.
It was shown that the dynamic characteristics of bolted joints are nonlinear and depends on mating interfaces and applied torque. Consequently, decreasing torque tend to increase frictional effect between mating interfaces and therefore increase damping in regions with less torque (1 Nm). Additionally, damping in lower frequency modes (modes 1 and 2) are more pronounced than in higher frequency mode (mode 3), in all studied case types for the 5-layer stack of 1.25 mm sheet steel elements.
Furthermore, there are uncertainties of how the number of bolts affects damping in layered sheet steel elements. However, based on experimental results it is evident that damping is dependent on the level of torque applied to a bolted joint. Therefore, an attempt to increase damping by decreasing tightening torque can affect the integrity of a bolted joint structure.
Three different (4, 8 and 12) rivet setups were studied for a 5-layer stack sheet steel elements, bound by riveted joints. Based on experimental results, high damping estimations were observed in setups with more riveted joints. Also, due to the presence of non-linear characteristics in riveted joints, the effects of material properties and joint stiffness on joint damping is still uncertain. Additionally, experimental results obtained for configurations with mechanically fastened joints and interfaces showed a considerable amount of irregular dynamic behaviors which were very challenging to account for in the numerical calculations.
It was shown that by utilizing the developed layered sheet steel modeling technique, the issues of nonlinearity generated from interacting mating surfaces can be resolved. To attest to this fact, the simulation results of the epoxy model is compared to the bolted stack, plastic
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ties and riveted stack models. From the correlated results the dynamic characteristics of the bolted stack, plastic ties and riveted stack were seen to be nonlinear and dependent on preload, clamping force, interacting interfaces and contact elements. Furthermore, by modeling the 5-layer stack of 1. 25 mm sheet steel as layered shell elements one is able to replicate a linear behavior for the layered stack since the developed technique does not consider friction between mating interfaces.
Additionally, it was observed, based on experimental results that, the dynamic response of the epoxy stack is linear. This is due to the close correlation between numerical and measured results. Therefore, based on the results gathered in this case studies, it can be concluded that, the most suitable way by which, the layered sheet steel elements in could be modeled in the lightweight wheel structure is by applying epoxy adhesive between the interacting surfaces of each layer. In this way the nonlinear effects due to mating interfaces are eliminated.
Numerical results from the dynamic analysis of the wheel structure revealed some normal vibration modes in the axial direction, meaning, should there be axial excitation forces, the dynamic performance of the wheel will be affected. However, the use of polymeric spacers as stiffeners for the layered steel sheets, will promote energy dissipation actions in all x, y, z global coordinate system which will consequently damp out all axial vibrations.
The proposed layered sheet steel modeling technique, led to a successful numerical analysis of the wheel structure. It was shown that, the vibrational modes of the stator, correlate well with those in literature for a cylindrical shell structure. Furthermore, though presented results were only limited to radial vibrations, the predicted and measured natural frequencies showed a very good correlation with a cr= 0.995.
The numerical analysis of the wheel, using the proposed layered sheet steel model, made provisions for eliminating mating interfaces for the layered sheet steel elements.
Additionally, since nonlinear effects from the bolted joints and mating interfaces were ignored, a linear dynamic analysis of the wheel structure was achieved. However, the dynamic characteristics of the quarter-scale wheel was observed to be nonlinear. This is as a result of the mating interfaces and frictional effects produced from the use of plastic ties and the bolted joints created by threaded rod cross tubes.
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The presented models for the layered sheet steel elements, needs to be further developed to accurately verify the FE models, since in these models, parameters such as clamping force, pretension and material properties are main contributors to the difference in predicted and measured data.
Also, further work should be carried to determine the effect of torsional vibrations on the prototype wheel. Furthermore, a numerical static structural analysis of the modified wheel structure should be investigated and results verified by experimental stress analysis of the prototype wheel structure, then later coupled to a harmonic analysis. All of which, will help to understand the harmonic behavior of the wheel structure.
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